If you want anyone to be able to create a case, you can set 'Allow public submissions' to 'Yes' in the edit page of a project. For anonymous users to be able to view the cases that they've created, they'll need to save the ticket that FogBugz provides them with when the case is created.
Otherwise, you can specify these parameters on a per-project basis. You can add community users to a project individually or by domain, or create groups of community users to add together to projects.
If you install the Community Case List plugin, you gain an enhanced interface for allowing community users to be able to view the list cases.
It sounds to me like what you most want is a publicly-submittable project augmented with the Community Case List plugin.